By Ehrenborg R.
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Additional resources for A Bijective Answer to a Question of Zvonkin
It is simpler to handle this quantity rather than the initial function, as the negative frequencies have been removed. Many authors use the time dependence exp(−j ωt), where the positive frequencies are removed. The complex representation of a wave travelling in the direction of positive x will be exp[j (k ∗ x − ωt)], because decreasing the amplitude in the direction of propagation implies that the new wave number is the complex conjugate of k. When adding the positive frequency components of a real signal to its negative frequency components, the initial real signal must be obtained.
2) where η is the shear viscosity (the volume viscosity is neglected), κ is the thermal conductivity, cp is the speciﬁc heat per unit mass at constant pressure. 6 10−2 w m−1 k−1 . The pressure is considered constant on a cross- section of the tube. 1, with a velocity υ(0, 0, υ3 ) parallel to the x3 axis and whose magnitude only depends on x1 . The variation of this velocity with x1 produces viscous stress. Due to the viscosity, the air is subjected to a shear force parallel to the x3 axis, and proportional to ∂υ3 /∂x1 .
The description of sound propagation in porous material can be complicated by the fact that sound also excites and moves the frame of the material. If the frame is motionless, in a ﬁrst step, the air inside the porous medium can be replaced on the macroscopic scale by an equivalent free ﬂuid. This equivalent ﬂuid has a complex effective density ρ and a complex bulk modulus K. The wave number k and the characteristic impedance Zc of the equivalent ﬂuid are also complex. 7, the porous layer can be replaced by a ﬂuid layer of density ρ/φ and of bulk modulus K/φ.